CORRESPONDENCE

2) Sawin and Maki point out that in the presence of several non-critical faults a fault-tolerant asynchronous network designed fromour technique may not mask critical faults. We have not distin-guished between critical and noncritical faults in deriving our re-sults and hence this comments is not directly applicable to our re-sults. An m-fault-tolerant network is guaranteed to mask only mfaults (both critical and noncritical) and may not mask more thanfaults (refer to the definition on p. 663 of our original paper).

3) Sawin and Maki observe that for some machines minimalstate variable designs may not necessarily require fewer gates thandesigns requiring a larger number of state variables. We believe thatthe comparison of reliabilities between different designs, made bySawin and Maki, is inconsistent with current technologies wheremodules more complex than gates are being used. Since the com-plexity and the probability of failure for the modules used in anactual realization cannot be generalized we believe that the re-liability measures on the basis of number of state variables is a viableone.

REFERENCES

[1] D. H. Sawin, III et al., "Design of asynchronous sequential machinesfor fault detection," in Dig. of 1972 Int. Symp. on Fault-TolerantComputing (Newton, Mass.), pp. 170-175, June 1972.

Comments on "Discrete Cosine Transform"K. SAM SHANMUGAM

Abstract-In the above correspondence,' Ahmed et al. proposed adiscrete cosine transform (DCT) and compared its performancewith the Karhunen-Loeve transform (KLT). They offered empiricalevidence showing that the DCT and the KLT compare closely for anumber of digital signal processing applications. The purpose ofthis note is to show that the DCT and the KLT are indeed asymptot-ically equivalent if the data covariance matrix is a Toeplitz matrix.

A number of orthogonal transforms are currently used in digitalimage processing and other digital signal processing applications.Recently, Ahmed et al.' defined a discrete cosine transform (DCT)and compared its performance with that of the Karhunen-Loevetransform (KLT) in image processing applications. They offeredempirical evidence showing that the DCT "compares closely" withthe performance of the KLT for a number of data processing applica-tions if the data covariance matrices are Toepliz matrices. Thepurpose of this comment is to point out that the DCT and the KLTare asymptotically equivalent for a class of Toeplitz matrices andhence it is not surprising that the performances of the DCT and theKLT are nearly identical for data vectors of reasonably large size.The proof to establish the asymptotic equivalence of the DCT and

KLT is based on some well-known results from the theory of circu-lant matrices and their relationship to Toeplitz matrices [1].

Consider the class of 2m X 2m Toeplitz matrices, T2m, whose(i,j)th entry, ti;, is defined as

tii = O

for i -jI < m - 1

O< p < 1

for i -jI > mi,j, = 0,1,2,* * *, (2m - 1).

(1)

Manuscript received March 1, 1974; revised July 29, 1974.The author is with the Department of Electrical Engineering, Wichita

State University, Wichita, Kans. 67208.1 N. Ahmed, T. Natarajan, and K. R. Rao, IEEE Trans. Comput.,

vol. C-23, pp. 90-93, Jan. 1974.

759

Such matrices are commonly used as models for the normalized datacovariance matrix for imagery data.

Also, consider a 2m X 2m circulant matrix C2., whose first row isgiven by,

(pO p .. . ,pml-, 0,pm-1l,pn,. . .p) (2)

The remaining rows of C2^ consist of (end around) cyclic shifts ofthe first row.The sequences of matrices T2m and C2m are asymptotically equiva-

lent and hence their eigenvalues are asymptotically equally dis-tributed. To show that T2. and C2. are asymptotically equivalent,we need to show that the T2m and C2m are uniformly bounded instrong norm, and T2m - C2- 0 in weak norm [1] as m-- oo.

For the T2m and C2i sequences defined in (1) and (2), both of theseconditions are satisfied by virtue of the fact that 0 < p < 1. Then,by Theorem 2.3 in Gray's paper, the eigenvalues of the two se-quences of matrices are asymptotically equally distributed.The eigenvalues, Xo,X1,,- ,X2m,- and the eigenvectors Vo,V1,V2,,

V2a"- of C2Oi are given by E1]2m-1 f27rik1

x = 2 Ck exp - | 2k=-0 m

(3)

where i = (-1)1/2 and ak are the entries in the first row of the cir-culant matrix C2m and

1I=~ {lww)(2 . . .,2mn-1l. c, x J27rijlV = (2m)"/2 co co = exp 2m .(4)

Equation (3) reduces to2in-I [2kjl

Xi = 2 CkCOs l 2m |k-0

(5)

since the ck, k = 0,1, -.,2m - 1 defined in (2) form an even se-quence. Hence, the eigenvalue distribution of the Toeplitz matrixT2m will be the same as (5) for large values of m.

If we use the mean-square error criterion for comparing the per-formance of transform techniques, then the KLT is optimum, andthe variance of the KLT samples are the eigenvalues of the datacovariance matrix. Any other orthogonal transform which providesthe same transform domain variance distribution as the KLT willperform as well as the KLT. Equation (5) shows that an appro-priately defined DCT will yield the same transform domain variancedistribution as the eigenvalues of the data covariance matrix andhence the performances of the DCT and the KLT will be comparablefor large values of m.

In view of this result, the empirical evidence presented in theabove mentioned paper' simply shows that even for small values of m,the performances of the DCT and the KLT are nearly identical.

REFERENCES[1] R. M. Gray, "On the asymptotic eigenvalue distribution of Toeplitz

matrices," IEEE Trans. Inform. Theory, vol. IT-18, pp. 725-730,Nov. 1972.

Comments on "Asynchronous Sequential MachinesDesigned for Fault Detection"

JOHN F. WAKERLY

Abstract-A recent paper' has proposed the design of asynchro-nous sequential machines with an internal fault detection mech-

Manuscript received May 20, 1974; revised August 20, 1974.The author is with the Digital Systems Laboratory, Stanford Uni-

versity, Stanford, Calif. 94305.1 D. H. Sawin, III, and G. K. Maki, IEEE Trans. Comput., vol. C-23,

pp. 239-249, Mar. 1974.